Question

The equation of the hyperbola whose foci are $(6, 5)$, $(- 4,5)$ and eccentricity $5/4$ is

• A. $\frac{\left(x-1\right)^{2}}{16}-\frac{\left(y-5\right)^{2}}{9}=1$
• B. $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$
• C. $\frac{\left(x-1\right)^{2}}{16}-\frac{\left(y-5\right)^{2}}{9}=-1$
• D. None of these

Answer 1

Answer: (A)

Centre of the hyperbola is the mid point of the line joining the two foci, therefore, the coordinates of the centre are $(1,5)$. Now distance between the foci $= 10$
$\Rightarrow 2ae = 10$
$ \Rightarrow ae = 5$
$\Rightarrow a = 4 \, \left[\because\, e=5 / 4\right]$
Now, $b^{2} = a^{2}\left(e^{2} - 1\right)$
$\Rightarrow b=3$
Hence, the equation of the hyperbola is
$\frac{\left(x-1\right)^{2}}{16}-\frac{\left(y-5\right)^{2}}{9}=1$.